Falkner and Boettcher, Equation (20:1,3)

Percentage Accurate: 99.3% → 99.5%

Time: 8.0s

Alternatives: 8

Speedup: 0.9×

Specification

Math

\[\begin{array}{l} \\ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \end{array} \]

FPCore

(FPCore (v t)
 :precision binary64
 (/
  (- 1.0 (* 5.0 (* v v)))
  (* (* (* (PI) t) (sqrt (* 2.0 (- 1.0 (* 3.0 (* v v)))))) (- 1.0 (* v v)))))

Initial Program: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \end{array} \]

FPCore

(FPCore (v t)
 :precision binary64
 (/
  (- 1.0 (* 5.0 (* v v)))
  (* (* (* (PI) t) (sqrt (* 2.0 (- 1.0 (* 3.0 (* v v)))))) (- 1.0 (* v v)))))

Alternative 1: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{t}}{\left(\left(\left(1 - v \cdot v\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{2}\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}} \end{array} \]

FPCore

(FPCore (v t)
 :precision binary64
 (/
  (/ (fma -5.0 (* v v) 1.0) t)
  (* (* (* (- 1.0 (* v v)) (PI)) (sqrt 2.0)) (sqrt (fma (* v v) -3.0 1.0)))))

Derivation

1. Initial program 99.4%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \cdot \sqrt{\frac{1}{1 - 3 \cdot {v}^{2}}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - 3 \cdot {v}^{2}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - 3 \cdot {v}^{2}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)}} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - 3 \cdot {v}^{2}}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   4. lower-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{1}{1 - 3 \cdot {v}^{2}}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   5. metadata-eval N/A

      \[\leadsto \sqrt{\frac{1}{1 - \color{blue}{\left(\mathsf{neg}\left(-3\right)\right)} \cdot {v}^{2}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   6. fp-cancel-sign-sub-inv N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{1 + -3 \cdot {v}^{2}}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   7. +-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{-3 \cdot {v}^{2} + 1}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(-3, {v}^{2}, 1\right)}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   9. unpow2 N/A

      \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(-3, \color{blue}{v \cdot v}, 1\right)}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(-3, \color{blue}{v \cdot v}, 1\right)}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   11. associate-/r* N/A

       \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(-3, v \cdot v, 1\right)}} \cdot \color{blue}{\frac{\frac{1 - 5 \cdot {v}^{2}}{t}}{\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)}} \]

   12. associate-/r* N/A

       \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(-3, v \cdot v, 1\right)}} \cdot \color{blue}{\frac{\frac{\frac{1 - 5 \cdot {v}^{2}}{t}}{\mathsf{PI}\left(\right)}}{\sqrt{2} \cdot \left(1 - {v}^{2}\right)}} \]

   13. lower-/.f64 N/A

       \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(-3, v \cdot v, 1\right)}} \cdot \color{blue}{\frac{\frac{\frac{1 - 5 \cdot {v}^{2}}{t}}{\mathsf{PI}\left(\right)}}{\sqrt{2} \cdot \left(1 - {v}^{2}\right)}} \]

5. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(-3, v \cdot v, 1\right)}} \cdot \frac{\frac{\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{t}}{\mathsf{PI}\left(\right)}}{\left(1 - v \cdot v\right) \cdot \sqrt{2}}} \]
6. Step-by-step derivation

7. Applied rewrites 99.6%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{t} \cdot 1}{\color{blue}{\left(\left(\left(1 - v \cdot v\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{2}\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}} \]

8. Final simplification 99.6%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{t}}{\left(\left(\left(1 - v \cdot v\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{2}\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}} \]
9. Add Preprocessing

Alternative 2: 98.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ {\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t\right)}^{-1} \end{array} \]

FPCore

(FPCore (v t) :precision binary64 (pow (* (* (sqrt 2.0) (PI)) t) -1.0))

Derivation

1. Initial program 99.4%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}} \]

   2. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right) \cdot t}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right) \cdot t}} \]

   4. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot t} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot t} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{\left(\color{blue}{\sqrt{2}} \cdot \mathsf{PI}\left(\right)\right) \cdot t} \]

   7. lower-PI.f64 99.0

      \[\leadsto \frac{1}{\left(\sqrt{2} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot t} \]

5. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\frac{1}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t}} \]

6. Final simplification 99.0%

   \[\leadsto {\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t\right)}^{-1} \]
7. Add Preprocessing

Alternative 3: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ {\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2}\right)}^{-1} \end{array} \]

FPCore

(FPCore (v t) :precision binary64 (pow (* (* (PI) t) (sqrt 2.0)) -1.0))

Derivation

1. Initial program 99.4%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}} \]

   2. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right) \cdot t}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right) \cdot t}} \]

   4. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot t} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot t} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{\left(\color{blue}{\sqrt{2}} \cdot \mathsf{PI}\left(\right)\right) \cdot t} \]

   7. lower-PI.f64 99.0

      \[\leadsto \frac{1}{\left(\sqrt{2} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot t} \]

5. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\frac{1}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t}} \]
6. Step-by-step derivation

7. Applied rewrites 98.9%

   \[\leadsto \frac{1}{\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \color{blue}{\sqrt{2}}} \]

8. Final simplification 98.9%

   \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2}\right)}^{-1} \]
9. Add Preprocessing

Alternative 4: 99.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(-2.5, v \cdot v, 1\right)}{\mathsf{PI}\left(\right)}}{\sqrt{2} \cdot t} \end{array} \]

FPCore

(FPCore (v t)
 :precision binary64
 (/ (/ (fma -2.5 (* v v) 1.0) (PI)) (* (sqrt 2.0) t)))

Derivation

1. Initial program 99.4%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \cdot \sqrt{\frac{1}{1 - 3 \cdot {v}^{2}}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - 3 \cdot {v}^{2}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - 3 \cdot {v}^{2}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)}} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - 3 \cdot {v}^{2}}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   4. lower-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{1}{1 - 3 \cdot {v}^{2}}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   5. metadata-eval N/A

      \[\leadsto \sqrt{\frac{1}{1 - \color{blue}{\left(\mathsf{neg}\left(-3\right)\right)} \cdot {v}^{2}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   6. fp-cancel-sign-sub-inv N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{1 + -3 \cdot {v}^{2}}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   7. +-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{-3 \cdot {v}^{2} + 1}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(-3, {v}^{2}, 1\right)}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   9. unpow2 N/A

      \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(-3, \color{blue}{v \cdot v}, 1\right)}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(-3, \color{blue}{v \cdot v}, 1\right)}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   11. associate-/r* N/A

       \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(-3, v \cdot v, 1\right)}} \cdot \color{blue}{\frac{\frac{1 - 5 \cdot {v}^{2}}{t}}{\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)}} \]

   12. associate-/r* N/A

       \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(-3, v \cdot v, 1\right)}} \cdot \color{blue}{\frac{\frac{\frac{1 - 5 \cdot {v}^{2}}{t}}{\mathsf{PI}\left(\right)}}{\sqrt{2} \cdot \left(1 - {v}^{2}\right)}} \]

   13. lower-/.f64 N/A

       \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(-3, v \cdot v, 1\right)}} \cdot \color{blue}{\frac{\frac{\frac{1 - 5 \cdot {v}^{2}}{t}}{\mathsf{PI}\left(\right)}}{\sqrt{2} \cdot \left(1 - {v}^{2}\right)}} \]

5. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(-3, v \cdot v, 1\right)}} \cdot \frac{\frac{\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{t}}{\mathsf{PI}\left(\right)}}{\left(1 - v \cdot v\right) \cdot \sqrt{2}}} \]

6. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{-5}{2} \cdot \frac{{v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)} + \frac{1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}} \]
7. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{-5}{2} \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}} + \frac{1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)} \]

   2. div-add-rev N/A

      \[\leadsto \color{blue}{\frac{\frac{-5}{2} \cdot {v}^{2} + 1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-5}{2} \cdot {v}^{2} + 1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\color{blue}{{v}^{2} \cdot \frac{-5}{2}} + 1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({v}^{2}, \frac{-5}{2}, 1\right)}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)} \]

   6. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{v \cdot v}, \frac{-5}{2}, 1\right)}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{v \cdot v}, \frac{-5}{2}, 1\right)}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)} \]

   8. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(v \cdot v, \frac{-5}{2}, 1\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right) \cdot t}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(v \cdot v, \frac{-5}{2}, 1\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right) \cdot t}} \]

   10. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(v \cdot v, \frac{-5}{2}, 1\right)}{\color{blue}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot t} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(v \cdot v, \frac{-5}{2}, 1\right)}{\color{blue}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot t} \]

   12. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(v \cdot v, \frac{-5}{2}, 1\right)}{\left(\color{blue}{\sqrt{2}} \cdot \mathsf{PI}\left(\right)\right) \cdot t} \]

   13. lower-PI.f64 99.3

       \[\leadsto \frac{\mathsf{fma}\left(v \cdot v, -2.5, 1\right)}{\left(\sqrt{2} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot t} \]

8. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(v \cdot v, -2.5, 1\right)}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t}} \]
9. Step-by-step derivation

10. Applied rewrites 99.4%

    \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-2.5, v \cdot v, 1\right)}{\mathsf{PI}\left(\right)}}{\sqrt{2} \cdot t}} \]
11. Add Preprocessing

Alternative 5: 99.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(-2.5, v \cdot v, 1\right)}{t}}{\sqrt{2} \cdot \mathsf{PI}\left(\right)} \end{array} \]

FPCore

(FPCore (v t)
 :precision binary64
 (/ (/ (fma -2.5 (* v v) 1.0) t) (* (sqrt 2.0) (PI))))

Derivation

1. Initial program 99.4%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \cdot \sqrt{\frac{1}{1 - 3 \cdot {v}^{2}}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - 3 \cdot {v}^{2}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - 3 \cdot {v}^{2}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)}} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - 3 \cdot {v}^{2}}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   4. lower-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{1}{1 - 3 \cdot {v}^{2}}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   5. metadata-eval N/A

      \[\leadsto \sqrt{\frac{1}{1 - \color{blue}{\left(\mathsf{neg}\left(-3\right)\right)} \cdot {v}^{2}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   6. fp-cancel-sign-sub-inv N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{1 + -3 \cdot {v}^{2}}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   7. +-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{-3 \cdot {v}^{2} + 1}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(-3, {v}^{2}, 1\right)}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   9. unpow2 N/A

      \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(-3, \color{blue}{v \cdot v}, 1\right)}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(-3, \color{blue}{v \cdot v}, 1\right)}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   11. associate-/r* N/A

       \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(-3, v \cdot v, 1\right)}} \cdot \color{blue}{\frac{\frac{1 - 5 \cdot {v}^{2}}{t}}{\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)}} \]

   12. associate-/r* N/A

       \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(-3, v \cdot v, 1\right)}} \cdot \color{blue}{\frac{\frac{\frac{1 - 5 \cdot {v}^{2}}{t}}{\mathsf{PI}\left(\right)}}{\sqrt{2} \cdot \left(1 - {v}^{2}\right)}} \]

   13. lower-/.f64 N/A

       \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(-3, v \cdot v, 1\right)}} \cdot \color{blue}{\frac{\frac{\frac{1 - 5 \cdot {v}^{2}}{t}}{\mathsf{PI}\left(\right)}}{\sqrt{2} \cdot \left(1 - {v}^{2}\right)}} \]

5. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(-3, v \cdot v, 1\right)}} \cdot \frac{\frac{\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{t}}{\mathsf{PI}\left(\right)}}{\left(1 - v \cdot v\right) \cdot \sqrt{2}}} \]

6. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{-5}{2} \cdot \frac{{v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)} + \frac{1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}} \]
7. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{-5}{2} \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}} + \frac{1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)} \]

   2. div-add-rev N/A

      \[\leadsto \color{blue}{\frac{\frac{-5}{2} \cdot {v}^{2} + 1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-5}{2} \cdot {v}^{2} + 1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\color{blue}{{v}^{2} \cdot \frac{-5}{2}} + 1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({v}^{2}, \frac{-5}{2}, 1\right)}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)} \]

   6. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{v \cdot v}, \frac{-5}{2}, 1\right)}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{v \cdot v}, \frac{-5}{2}, 1\right)}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)} \]

   8. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(v \cdot v, \frac{-5}{2}, 1\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right) \cdot t}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(v \cdot v, \frac{-5}{2}, 1\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right) \cdot t}} \]

   10. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(v \cdot v, \frac{-5}{2}, 1\right)}{\color{blue}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot t} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(v \cdot v, \frac{-5}{2}, 1\right)}{\color{blue}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot t} \]

   12. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(v \cdot v, \frac{-5}{2}, 1\right)}{\left(\color{blue}{\sqrt{2}} \cdot \mathsf{PI}\left(\right)\right) \cdot t} \]

   13. lower-PI.f64 99.3

       \[\leadsto \frac{\mathsf{fma}\left(v \cdot v, -2.5, 1\right)}{\left(\sqrt{2} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot t} \]

8. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(v \cdot v, -2.5, 1\right)}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t}} \]
9. Step-by-step derivation

10. Applied rewrites 99.4%

    \[\leadsto \frac{\frac{\mathsf{fma}\left(-2.5, v \cdot v, 1\right)}{t}}{\color{blue}{\sqrt{2} \cdot \mathsf{PI}\left(\right)}} \]
11. Add Preprocessing

Alternative 6: 98.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-2.5, v \cdot v, 1\right)}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t} \end{array} \]

FPCore

(FPCore (v t)
 :precision binary64
 (/ (fma -2.5 (* v v) 1.0) (* (* (sqrt 2.0) (PI)) t)))

Derivation

1. Initial program 99.4%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{-5}{2} \cdot \frac{{v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)} + \frac{1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{-5}{2} \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}} + \frac{1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)} \]

   2. div-add-rev N/A

      \[\leadsto \color{blue}{\frac{\frac{-5}{2} \cdot {v}^{2} + 1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-5}{2} \cdot {v}^{2} + 1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-5}{2}, {v}^{2}, 1\right)}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)} \]

   5. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-5}{2}, \color{blue}{v \cdot v}, 1\right)}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-5}{2}, \color{blue}{v \cdot v}, 1\right)}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-5}{2}, v \cdot v, 1\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right) \cdot t}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-5}{2}, v \cdot v, 1\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right) \cdot t}} \]

   9. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-5}{2}, v \cdot v, 1\right)}{\color{blue}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot t} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-5}{2}, v \cdot v, 1\right)}{\color{blue}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot t} \]

   11. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-5}{2}, v \cdot v, 1\right)}{\left(\color{blue}{\sqrt{2}} \cdot \mathsf{PI}\left(\right)\right) \cdot t} \]

   12. lower-PI.f64 99.3

       \[\leadsto \frac{\mathsf{fma}\left(-2.5, v \cdot v, 1\right)}{\left(\sqrt{2} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot t} \]

5. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2.5, v \cdot v, 1\right)}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t}} \]
6. Add Preprocessing

Alternative 7: 98.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\mathsf{PI}\left(\right)}}{\sqrt{2} \cdot t} \end{array} \]

FPCore

(FPCore (v t) :precision binary64 (/ (/ 1.0 (PI)) (* (sqrt 2.0) t)))

Derivation

1. Initial program 99.4%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \cdot \sqrt{\frac{1}{1 - 3 \cdot {v}^{2}}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - 3 \cdot {v}^{2}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - 3 \cdot {v}^{2}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)}} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - 3 \cdot {v}^{2}}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   4. lower-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{1}{1 - 3 \cdot {v}^{2}}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   5. metadata-eval N/A

      \[\leadsto \sqrt{\frac{1}{1 - \color{blue}{\left(\mathsf{neg}\left(-3\right)\right)} \cdot {v}^{2}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   6. fp-cancel-sign-sub-inv N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{1 + -3 \cdot {v}^{2}}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   7. +-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{-3 \cdot {v}^{2} + 1}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(-3, {v}^{2}, 1\right)}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   9. unpow2 N/A

      \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(-3, \color{blue}{v \cdot v}, 1\right)}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(-3, \color{blue}{v \cdot v}, 1\right)}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   11. associate-/r* N/A

       \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(-3, v \cdot v, 1\right)}} \cdot \color{blue}{\frac{\frac{1 - 5 \cdot {v}^{2}}{t}}{\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)}} \]

   12. associate-/r* N/A

       \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(-3, v \cdot v, 1\right)}} \cdot \color{blue}{\frac{\frac{\frac{1 - 5 \cdot {v}^{2}}{t}}{\mathsf{PI}\left(\right)}}{\sqrt{2} \cdot \left(1 - {v}^{2}\right)}} \]

   13. lower-/.f64 N/A

       \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(-3, v \cdot v, 1\right)}} \cdot \color{blue}{\frac{\frac{\frac{1 - 5 \cdot {v}^{2}}{t}}{\mathsf{PI}\left(\right)}}{\sqrt{2} \cdot \left(1 - {v}^{2}\right)}} \]

5. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(-3, v \cdot v, 1\right)}} \cdot \frac{\frac{\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{t}}{\mathsf{PI}\left(\right)}}{\left(1 - v \cdot v\right) \cdot \sqrt{2}}} \]

6. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{-5}{2} \cdot \frac{{v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)} + \frac{1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}} \]
7. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{-5}{2} \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}} + \frac{1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)} \]

   2. div-add-rev N/A

      \[\leadsto \color{blue}{\frac{\frac{-5}{2} \cdot {v}^{2} + 1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-5}{2} \cdot {v}^{2} + 1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\color{blue}{{v}^{2} \cdot \frac{-5}{2}} + 1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({v}^{2}, \frac{-5}{2}, 1\right)}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)} \]

   6. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{v \cdot v}, \frac{-5}{2}, 1\right)}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{v \cdot v}, \frac{-5}{2}, 1\right)}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)} \]

   8. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(v \cdot v, \frac{-5}{2}, 1\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right) \cdot t}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(v \cdot v, \frac{-5}{2}, 1\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right) \cdot t}} \]

   10. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(v \cdot v, \frac{-5}{2}, 1\right)}{\color{blue}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot t} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(v \cdot v, \frac{-5}{2}, 1\right)}{\color{blue}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot t} \]

   12. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(v \cdot v, \frac{-5}{2}, 1\right)}{\left(\color{blue}{\sqrt{2}} \cdot \mathsf{PI}\left(\right)\right) \cdot t} \]

   13. lower-PI.f64 99.3

       \[\leadsto \frac{\mathsf{fma}\left(v \cdot v, -2.5, 1\right)}{\left(\sqrt{2} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot t} \]

8. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(v \cdot v, -2.5, 1\right)}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t}} \]
9. Step-by-step derivation

10. Applied rewrites 99.4%

    \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-2.5, v \cdot v, 1\right)}{\mathsf{PI}\left(\right)}}{\sqrt{2} \cdot t}} \]

11. Taylor expanded in v around 0

    \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{\sqrt{\color{blue}{2}} \cdot t} \]
12. Step-by-step derivation

13. Applied rewrites 99.1%

    \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{\sqrt{\color{blue}{2}} \cdot t} \]
14. Add Preprocessing

Alternative 8: 98.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{t}}{\sqrt{2} \cdot \mathsf{PI}\left(\right)} \end{array} \]

FPCore

(FPCore (v t) :precision binary64 (/ (/ 1.0 t) (* (sqrt 2.0) (PI))))

Derivation

1. Initial program 99.4%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \cdot \sqrt{\frac{1}{1 - 3 \cdot {v}^{2}}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - 3 \cdot {v}^{2}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - 3 \cdot {v}^{2}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)}} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - 3 \cdot {v}^{2}}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   4. lower-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{1}{1 - 3 \cdot {v}^{2}}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   5. metadata-eval N/A

      \[\leadsto \sqrt{\frac{1}{1 - \color{blue}{\left(\mathsf{neg}\left(-3\right)\right)} \cdot {v}^{2}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   6. fp-cancel-sign-sub-inv N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{1 + -3 \cdot {v}^{2}}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   7. +-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{-3 \cdot {v}^{2} + 1}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(-3, {v}^{2}, 1\right)}}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   9. unpow2 N/A

      \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(-3, \color{blue}{v \cdot v}, 1\right)}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(-3, \color{blue}{v \cdot v}, 1\right)}} \cdot \frac{1 - 5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)\right)} \]

   11. associate-/r* N/A

       \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(-3, v \cdot v, 1\right)}} \cdot \color{blue}{\frac{\frac{1 - 5 \cdot {v}^{2}}{t}}{\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \left(1 - {v}^{2}\right)\right)}} \]

   12. associate-/r* N/A

       \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(-3, v \cdot v, 1\right)}} \cdot \color{blue}{\frac{\frac{\frac{1 - 5 \cdot {v}^{2}}{t}}{\mathsf{PI}\left(\right)}}{\sqrt{2} \cdot \left(1 - {v}^{2}\right)}} \]

   13. lower-/.f64 N/A

       \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(-3, v \cdot v, 1\right)}} \cdot \color{blue}{\frac{\frac{\frac{1 - 5 \cdot {v}^{2}}{t}}{\mathsf{PI}\left(\right)}}{\sqrt{2} \cdot \left(1 - {v}^{2}\right)}} \]

5. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(-3, v \cdot v, 1\right)}} \cdot \frac{\frac{\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{t}}{\mathsf{PI}\left(\right)}}{\left(1 - v \cdot v\right) \cdot \sqrt{2}}} \]

6. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{-5}{2} \cdot \frac{{v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)} + \frac{1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}} \]
7. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{-5}{2} \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}} + \frac{1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)} \]

   2. div-add-rev N/A

      \[\leadsto \color{blue}{\frac{\frac{-5}{2} \cdot {v}^{2} + 1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-5}{2} \cdot {v}^{2} + 1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\color{blue}{{v}^{2} \cdot \frac{-5}{2}} + 1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({v}^{2}, \frac{-5}{2}, 1\right)}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)} \]

   6. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{v \cdot v}, \frac{-5}{2}, 1\right)}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{v \cdot v}, \frac{-5}{2}, 1\right)}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)} \]

   8. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(v \cdot v, \frac{-5}{2}, 1\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right) \cdot t}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(v \cdot v, \frac{-5}{2}, 1\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right) \cdot t}} \]

   10. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(v \cdot v, \frac{-5}{2}, 1\right)}{\color{blue}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot t} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(v \cdot v, \frac{-5}{2}, 1\right)}{\color{blue}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot t} \]

   12. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(v \cdot v, \frac{-5}{2}, 1\right)}{\left(\color{blue}{\sqrt{2}} \cdot \mathsf{PI}\left(\right)\right) \cdot t} \]

   13. lower-PI.f64 99.3

       \[\leadsto \frac{\mathsf{fma}\left(v \cdot v, -2.5, 1\right)}{\left(\sqrt{2} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot t} \]

8. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(v \cdot v, -2.5, 1\right)}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t}} \]
9. Step-by-step derivation

10. Applied rewrites 99.4%

    \[\leadsto \frac{\frac{\mathsf{fma}\left(-2.5, v \cdot v, 1\right)}{t}}{\color{blue}{\sqrt{2} \cdot \mathsf{PI}\left(\right)}} \]

11. Taylor expanded in v around 0

    \[\leadsto \frac{\frac{1}{t}}{\sqrt{\color{blue}{2}} \cdot \mathsf{PI}\left(\right)} \]
12. Step-by-step derivation

13. Applied rewrites 99.1%

    \[\leadsto \frac{\frac{1}{t}}{\sqrt{\color{blue}{2}} \cdot \mathsf{PI}\left(\right)} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024326 
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  :precision binary64
  (/ (- 1.0 (* 5.0 (* v v))) (* (* (* (PI) t) (sqrt (* 2.0 (- 1.0 (* 3.0 (* v v)))))) (- 1.0 (* v v)))))